arXiv subjects
Publications and source records attributed to Juan Carlos Marrero.
We construct the exponential map associated to a nonholonomic system that allows us to define an exact discrete nonholonomic constraint submanifold. We reproduce the continuous nonholonomic flow as a discrete flow on this discrete constraint submanifold deriving an exact discrete version of the nonholonomic equations. Finally, we derive a general family of nonholonomic integrators.
We provide models that are as close as possible to being formal for a large class of compact manifolds that admit a transversely Kaehler structure, including Vaisman and quasi-Sasakian manifolds. As an application we are able to classify the corresponding nilmanifolds.
In this note we discuss symplectic lifts of actions for a complete Lagrangian fibration. Firstly, we describe the symplectic cotangent lifts of a G-action on a manifold Q in terms of 1-cocycles in the cohomology of G induced by the action with values in the space of closed 1-forms on Q. After this, we consider the general case of complete Lagrangian fibrations.
In this paper we develop a geometric version of the Hamilton-Jacobi equation in the Poisson setting. Specifically, we "geometrize" what is usually called a complete solution of the Hamilton-Jacobi equation. We use some well-known results about symplectic groupoids, in particular cotangent groupoids, as a keystone for the construction of our framework. Our methodology follows the ambitious program proposed by A. Weinstein, [62], in order to develop geometric formulations of the dynamical behavior of Lagrangian and Hamiltonian systems on Lie algebroids and Lie groupoids. This procedure allows us to take symmetries into account, and, as a by-product, we recover results from [14, 29, 31], but even in these situations our approach is new. A theory of generating functions for the Poisson structures considered here is also developed following the same pattern, solving a longstanding problem of the area: how to obtain a generating function for the identity tranformation and the nearby Poisson automorphisms of Poisson manifolds. A direct application of our results give the construction of a family of Poisson integrators, that is, integrators that conserve the underlying Poisson geometry. These integrators are implemented in the paper in benchmark problems. Some conclusions, current and future directions of research are shown at the end of the paper.
Given a Lie-Poisson completely integrable bi-Hamiltonian system on $\mathbb{R}^n$, we present a method which allows us to construct, under certain conditions, a completely integrable bi-Hamiltonian deformation of the initial Lie-Poisson system on a non-abelian Poisson-Lie group $G_η$ of dimension $n$, where $η\in \mathbb{R}$ is the deformation parameter. Moreover, we show that from the two multiplicative (Poisson-Lie) Hamiltonian structures on $G_η$ that underly the dynamics of the deformed system and by making use of the group law on $G_η$, one may obtain two completely integrable Hamiltonian systems on $G_η\times G_η$. By construction, both systems admit reduction, via the multiplication in $G_η$, to the deformed bi-Hamiltonian system in $G_η$. The previous approach is applied to two relevant Lie-Poisson completely integrable bi-Hamiltonian systems: the Lorenz and Euler top systems.
We establish a Hard Lefschetz Theorem for the de Rham cohomology of compact Vaisman manifolds. A similar result is proved for the basic cohomology with respect to the Lee vector field. Motivated by these results, we introduce the notions of a Lefschetz and of a basic Lefschetz locally conformal symplectic (l.c.s.) manifold of the first kind. We prove that the two notions are equivalent if there exists a Riemannian metric such that the Lee vector field is unitary and parallel and its metric dual $1$-form coincides with the Lee $1$-form. Finally, we discuss several examples of compact l.c.s. manifolds of the first kind which do not admit compatible Vaisman metrics.
We extend to metric compact mapping tori a splitting result for coKähler manifolds. In particular, we prove that a compact Vaisman manifold is finitely covered by the product of a Sasakian manifold and a circle.
We present some examples of locally conformal symplectic structures of the first kind on compact nilmanifolds which do not admit Vaisman metrics. One of these examples does not admit locally conformal Kähler metrics and all the structures come from left-invariant locally conformal symplectic structures on the corresponding nilpotent Lie groups. Under certain topological restrictions related with the compactness of the canonical foliation, we prove a structure theorem for locally conformal symplectic manifolds of the first kind. In the non compact case, we show that they are the product of a real line with a compact contact manifold and, in the compact case, we obtain that they are mapping tori of compact contact manifolds by strict contactomorphisms. Motivated by the aforementioned examples, we also study left-invariant locally conformal symplectic structures on Lie groups. In particular, we obtain a complete description of these structures (with non-zero Lee $1$-form) on connected simply connected nilpotent Lie groups in terms of locally conformal symplectic extensions and symplectic double extensions of symplectic nilpotent Lie groups. In order to obtain this description, we study locally conformal symplectic structures of the first kind on Lie algebras.
We find a family of five dimensional completely solvable compact manifolds that constitute the first examples of $K$-contact manifolds which satisfy the Hard Lefschetz Theorem and have a model of Tievsky type just as Sasakian manifolds but do not admit any Sasakian structure.
In this article, we generalize the theory of discrete Lagrangian mechanics and variational integrators in two principal directions. First, we show that Lagrangian submanifolds of symplectic groupoids give rise to discrete dynamical systems, and we study the properties of these systems, including their regularity and reversibility, from the perspective of symplectic and Poisson geometry. Next, we use this framework -- along with a generalized notion of generating function due to Sniatycki and Tulczyjew -- to develop a theory of discrete constrained Lagrangian mechanics. This allows for systems with arbitrary constraints, including those which are non-integrable (in an appropriate discrete, variational sense). In addition to characterizing the dynamics of these constrained systems, we also develop a theory of reduction and Noether symmetries, and study the relationship between the dynamics and variational principles. Finally, we apply this theory to discretize several concrete examples of constrained systems in mechanics and optimal control.
The aim of this paper is to generalize the classical Marsden-Weinstein reduction procedure for symplectic manifolds to polysymplectic manifolds in order to obtain quotient manifolds which in- herit the polysymplectic structure. This generalization allows us to reduce polysymplectic Hamiltonian systems with symmetries, such as those appearing in certain kinds of classical field theories. As an application of this technique, an analogous to the Kirillov-Kostant-Souriau theorem for polysymplectic manifolds is obtained and some other mathematical examples are also analyzed. Our procedure corrects some mistakes and inaccuracies in previous papers [29, 50] on this subject.
We prove that a compact nilmanifold admits a Sasakian structure if and only if it is a compact quotient of the generalized Heisenberg group of odd dimension by a co-compact discrete subgroup.
Using the Hard Lefschetz Theorem for Sasakian manifolds, we find two examples of compact K-contact nilmanifolds with no compatible Sasakian metric in dimensions five and seven, respectively
We obtain an example of a compact locally conformal symplectic nilmanifold which admits no locally conformal Kähler metrics. This gives a new positive answer to a question raised by L. Ornea and M. Verbitsky.
In this paper, we introduce local expressions for discrete Mechanics. To apply our results simultaneously to several interesting cases, we derive these local expressions in the framework of Lie groupoids, following the program proposed by Alan Weinstein in [19]. To do this, we will need some results on the geometry of Lie groupoids, as, for instance, the construction of symmetric neighborhoods or the existence of local bisections. These local descriptions will be particular useful for the explicit construction of geometric integrators for mechanical systems (reduced or not), in particular, discrete Euler-Lagrange equations, discrete Euler-Poincaré equations, discrete Lagrange-Poincaré equations... The results contained in this paper can be considered as a local version of the study that we have started in [13], on the geometry of discrete Mechanics on Lie groupoids.
We describe a reduction process for symplectic principal $\mathbb{R}$-bundles in the presence of a momentum map. This type of structures plays an important role in the geometric formulation of non-autonomous Hamiltonian systems. We apply this procedure to the standard symplectic principal $\mathbb{R}$-bundle associated with a fibration $π:M\to\mathbb{R}$. When $π$ is a principal $G$-bundle and $G_ν$ denotes the isotropy group associated with an element $ν$ in the dual to the Lie algebra of $G$, we use the reduction process in order to describe a Poisson structure on the quotient manifold $M/G_ν$ whose symplectic leaves are isomorphic to the coadjoint orbit $\mathcal{O}_ν$ . Moreover, we show a reduction process for non-autonomous Hamiltonian systems on symplectic principal $\mathbb{R}$-bundles.
We develop a Hamilton-Jacobi theory for singular lagrangian systems using the Gotay-Nester-Hinds constraint algorithm. The procedure works even if the system has secondary constraints.
This article addresses the problem of developing an extension of the Marsden- Weinstein reduction process to symplectic Lie algebroids, and in particular to the case of the symplectic cover of a fiberwise linear Poisson structure, whose reduction process is the analogue to cotangent bundle reduction in the context of Lie algebroids.